Allee’s Effect Bifurcation in Generalized Logistic Maps
Abstract
This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.
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The Logistic map exhibits complex dynamical behavior when the parameter p is in the interval (3.57, 4].
“The Logistic map exhibits complex dynamical behavior when the parameter p is in the interval (3.57, 4].”
From Chaos-Based Image Encryption: Review, Application, and Challenges (2023) · cites this paper as [15]
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Chaos-Based Image Encryption: Review, Application, and Challenges
Zhang, B., & Liu, L. (2023) · Mathematics · doi:10.3390/math11112585 · Cited by 328 · see citing works
“Several examples of discrete chaotic maps and their mathematical definitions are described below to provide a better understanding: Logistic map [15]: where X t ∈ (0, 1) is the state variable at time step t , p ∈ [0, 4] is a control parameter, and (1 − X t ) is the factor that limits the growth of…”
Background
- Authors
- Leonel Rocha, J., & Taha, A.-K.
- Affiliations
- CEAUL. ADM, ISEL-Engineering Superior Institute of Lisbon, Polytechnic Institute of Lisbon, Rua Conselheiro Emídio Navarro 1, 1959-007 Lisboa, Portugal
- Université Fédérale de Toulouse Midi-Pyrénées (FR)
- ORCID
- Taha, Abdel-Kaddous
- Venue
- International Journal of Bifurcation and Chaos
- Publisher
- World Scientific Pub Co Pte Ltd
- Published
- 2019-03
- Volume
- 29
- Issue
- 03
- Pages
- 1950039
- ISSN
- 0218-1274 (ISSN-L)
- 1793-6551
- OpenAlex
- W2929444604
- Cited by
- 43
- References
- 0
- Citations by year
- 2025: 11 · 2024: 12 · 2023: 12 · 2022: 3 · 2021: 2 · 2020: 2 · 2019: 1
- FWCI
- 2.39
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- International Journal of Bifurcation and Chaos (publishedVersion)
- Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT) (submittedVersion)
- Funders
- Fundação para a Ciência e Tecnologia (UID/MAT/00006/2013)
- Topics
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
- Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Equations and Dynamical Systems
- Keywords
- Allee effect, one-dimensional maps, unimodal maps, snap-back repeller, fixed point stability, bifurcation, extinction, parameter space, global bifurcation
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